Difference polynomials

In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and the Stirling interpolation polynomials as special cases.

Definition

The general difference polynomial sequence is given by

p_n(z)=\frac{z}{n} {{z-\beta n -1} \choose {n-1}}

where {z \choose n} is the binomial coefficient. For \beta=0, the generated polynomials p_n(z) are the Newton polynomials

p_n(z)= {z \choose n} = \frac{z(z-1)\cdots(z-n+1)}{n!}.

The case of \beta=1 generates Selberg's polynomials, and the case of \beta=-1/2 generates Stirling's interpolation polynomials.

Moving differences

Given an analytic function f(z), define the moving difference of f as

\mathcal{L}_n(f) = \Delta^n f (\beta n)

where \Delta is the forward difference operator. Then, provided that f obeys certain summability conditions, then it may be represented in terms of these polynomials as

f(z)=\sum_{n=0}^\infty p_n(z) \mathcal{L}_n(f).

The conditions for summability (that is, convergence) for this sequence is a fairly complex topic; in general, one may say that a necessary condition is that the analytic function be of less than exponential type. Summability conditions are discussed in detail in Boas & Buck.

Generating function

The generating function for the general difference polynomials is given by

e^{zt}=\sum_{n=0}^\infty p_n(z) \left[\left(e^t-1\right)e^{\beta t}\right]^n.

This generating function can be brought into the form of the generalized Appell representation

K(z,w) = A(w)\Psi(zg(w)) = \sum_{n=0}^\infty p_n(z) w^n

by setting A(w)=1, \Psi(x)=e^x, g(w)=t and w=(e^t-1)e^{\beta t}.

See also

References